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In this paper, we study the geometric decomposition of the degree-$0$ Persistent Homology Transform (PHT) as viewed as a persistence diagram bundle. We focus on star-shaped objects as they can be segmented into smaller, simpler regions known as ``sectors''. Algebraically, we demonstrate that the degree-$0$ persistence diagram of a star-shaped object in $\mathbb{R}^2$ can be derived from the degree-$0$ persistence diagrams of its sectors. Using this, we then establish sufficient conditions for star-shaped objects in $\mathbb{R}^2$ so that they have ``trivial geometric monodromy''. Consequently, the PHT of such a shape can be decomposed as a union of curves parameterized by $S^1$, where the curves are given by the continuous movement of each point in the persistence diagrams that are parameterized by $S^{1}$. Finally, we discuss the current challenges of generalizing these results to higher dimensions., Comment: 26 pages, 6 Figures

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, NervePool, for data structured as simplicial complexes, which are generalizations of graphs that include higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. The proposed simplicial coarsening scheme is built upon partitions of vertices, which allow us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. NervePool builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice, the pooling operations are computed via a series of matrix operations, the topological motivation is a set-theoretic construction based on unions of stars of simplices and the nerve complex, Comment: 22 pages, 9 figures

There is a growing body of work that leverages features extracted via topological data analysis to train machine learning models. While this field, sometimes known as topological machine learning (TML), has seen some notable successes, an understanding of how the process of learning from topological features differs from the process of learning from raw data is still limited. In this work, we begin to address one component of this larger issue by asking whether a model trained with topological features learns internal representations of data that are fundamentally different than those learned by a model trained with the original raw data. To quantify ``different'', we exploit two popular metrics that can be used to measure the similarity of the hidden representations of data within neural networks, neural stitching and centered kernel alignment. From these we draw a range of conclusions about how training with topological features does and does not change the representations that a model learns. Perhaps unsurprisingly, we find that structurally, the hidden representations of models trained and evaluated on topological features differ substantially compared to those trained and evaluated on the corresponding raw data. On the other hand, our experiments show that in some cases, these representations can be reconciled (at least to the degree required to solve the corresponding task) using a simple affine transformation. We conjecture that this means that neural networks trained on raw data may extract some limited topological features in the process of making predictions., Comment: To appear at NeurIPS 2022 Workshop on Symmetry and Geometry in Neural Representations (NeurReps)

Graph Neural Networks (GNNs) extend the success of neural networks to graph-structured data by accounting for their intrinsic geometry. While extensive research has been done on developing GNN models with superior performance according to a collection of graph representation learning benchmarks, it is currently not well understood what aspects of a given model are probed by them. For example, to what extent do they test the ability of a model to leverage graph structure vs. node features? Here, we develop a principled approach to taxonomize benchmarking datasets according to a $\textit{sensitivity profile}$ that is based on how much GNN performance changes due to a collection of graph perturbations. Our data-driven analysis provides a deeper understanding of which benchmarking data characteristics are leveraged by GNNs. Consequently, our taxonomy can aid in selection and development of adequate graph benchmarks, and better informed evaluation of future GNN methods. Finally, our approach and implementation in $\texttt{GTaxoGym}$ package are extendable to multiple graph prediction task types and future datasets., Comment: In Proceedings of the First Learning on Graphs Conference (LoG 2022)

The Internet is transforming our society, necessitating a quantitative understanding of Internet traffic. Our team collects and curates the largest publicly available Internet traffic data sets. An analysis of 50 billion packets using 10,000 processors in the MIT SuperCloud reveals a new phenomenon: the importance of otherwise unseen leaf nodes and isolated links in Internet traffic. Our analysis further shows that a two-parameter modified Zipf-Mandelbrot distribution accurately describes a wide variety of source/destination statistics on moving sample windows ranging from 100{,}000 to 100{,}000{,}000 packets over collections that span years and continents. The measured model parameters distinguish different network streams, and the model leaf parameter strongly correlates with the fraction of the traffic in different underlying network topologies., Comment: 53 pages, 27 figures, 8 tables, 121 references. Portions of this work originally appeared as arXiv:1904.04396v1 which has been split for publication in the book "Massive Graph Analytics" (edited by David Bader)

Graph neural networks (GNNs) have attracted much attention due to their ability to leverage the intrinsic geometries of the underlying data. Although many different types of GNN models have been developed, with many benchmarking procedures to demonstrate the superiority of one GNN model over the others, there is a lack of systematic understanding of the underlying benchmarking datasets, and what aspects of the model are being tested. Here, we provide a principled approach to taxonomize graph benchmarking datasets by carefully designing a collection of graph perturbations to probe the essential data characteristics that GNN models leverage to perform predictions. Our data-driven taxonomization of graph datasets provides a new understanding of critical dataset characteristics that will enable better model evaluation and the development of more specialized GNN models., Comment: in Data-Centric AI Workshop at NeurIPS 2021

Our society has never been more dependent on computer networks. Effective utilization of networks requires a detailed understanding of the normal background behaviors of network traffic. Large-scale measurements of networks are computationally challenging. Building on prior work in interactive supercomputing and GraphBLAS hypersparse hierarchical traffic matrices, we have developed an efficient method for computing a wide variety of streaming network quantities on diverse time scales. Applying these methods to 100,000,000,000 anonymized source-destination pairs collected at a network gateway reveals many previously unobserved scaling relationships. These observations provide new insights into normal network background traffic that could be used for anomaly detection, AI feature engineering, and testing theoretical models of streaming networks., Comment: 6 pages, 6 figures,3 tables, 49 references, accepted to IEEE HPEC 2020

Purpose: This study assessed the dosimetric accuracy of synthetic CT images generated from magnetic resonance imaging (MRI) data for focal brain radiation therapy, using a deep learning approach. Material and Methods: We conducted a study in 77 patients with brain tumors who had undergone both MRI and computed tomography (CT) imaging as part of their simulation for external beam treatment planning. We designed a generative adversarial network (GAN) to generate synthetic CT images from MRI images. We used Mutual Information (MI) as the loss function in the generator to overcome the misalignment between MRI and CT images (unregistered data). The model was trained using all MRI slices with corresponding CT slices from each training subject s MRI/CT pair. Results: The proposed GAN method produced an average mean absolute error (MAE) of 47.2 +- 11.0 HU over 5-fold cross validation. The overall mean Dice similarity coefficient between CT and synthetic CT images was 80% +- 6% in bone for all test data. Though training a GAN model may take several hours, the model only needs to be trained once. Generating a complete synthetic CT volume for each new patient MRI volume using a trained GAN model took only one second. Conclusions: The GAN model we developed produced highly accurate synthetic CT images from conventional, single-sequence MRI images in seconds. Our proposed method has strong potential to perform well in a clinical workflow for MRI-only brain treatment planning.

Inter-and intra-observer variation in delineating regions of interest (ROIs) occurs because of differences in expertise level and preferences of the radiation oncologists. We evaluated the accuracy of a segmentation model using the U-Net structure to delineate the prostate, bladder, and rectum in male pelvic CT images. The dataset used for training and testing the model consisted of raw CT scan images of 85 prostate cancer patients. We designed a 2D U-Net model to directly learn a mapping function that converts a 2D CT grayscale image to its corresponding 2D OAR segmented image. Our network contains blocks of convolution 2D layers with variable kernel sizes, channel number, and activation functions. On the left side of the U-Net model, we used three 3x3 convolutions, each followed by a rectified linear unit (ReLu) (activation function), and one max pooling operation. On the right side of the U-Net model, we used a 2x2 transposed convolution and two 3x3 convolution networks followed by a ReLu activation function. The automatic segmentation using the U-Net generated an average dice similarity coefficient (DC) and standard deviation (SD) of the following: DC +- SD (0.88 +- 0.12), (0.95 +- 0.04), and (0.92 +- 0.06) for the prostate, bladder, and rectum, respectively. Furthermore, the mean of average surface Hausdorff distance (ASHD) and SD were 1.2 +- 0.9 mm, 1.08 +- 0.8 mm, and 0.8 +- 0.6 mm for the prostate, bladder, and rectum, respectively. Our proposed method, which employs the U-Net structure, is highly accurate and reproducible for automated ROI segmentation. This provides a foundation to improve automatic delineation of the boundaries between the target and surrounding normal soft tissues on a standard radiation therapy planning CT scan., Comment: 18 pages, 4 figures, 3 tables